The σ-Pair Observable
Within V600 = 2I, the σ-Galois involution partitions the 120 vertices into 24 fixed and 96 mobile (Paper 1). The mobile set carries a natural pairing under σ: each mobile vertex v sits opposite its σ-image σ(v), forming 48 unordered pairs. The σ-pair excitation observable measures the discrete energy carried by exciting one such pair.
96 = 6 cosets × 16
From the Paper-1 incidence theorem, each of the 6 Dic5-cosets contains 16 mobile vertices, summing to 96 mobile vertices in total across V600.
48 disjoint pairs
The mobile set is partitioned by σ into 48 unordered pairs {v, σ(v)}, each contained inside one Dic5-coset by the cosets' σ-stability.
The Spectrum — Monochromatic
The proof: write the observable in the basis of mobile vertices, evaluate the matrix element on each σ-pair, and use the Dic5-coset action to identify the value across cosets. The 5/2 emerges from the icosian trace structure inherited from V600 — not as a free parameter, but as a forced ratio in the underlying ring ℤ[φ].
The Cascading First-Law Identity
Within each coset, the count of fixed vertices (4) and the count of mobile vertices (16) combine with the monochromatic energy Eq = 5/2 to yield a closed identity in which the same arithmetic returns under variations of fixed-count or mobile-count:
The identity is explicitly described as a discrete analogue of the black-hole first law, not as a derivation of Hawking's continuum result. The paper offers a semiclassical-recovery sketch — how the discrete arithmetic might project onto Hawking's continuum spectrum under a stated cutoff — but does not complete it. Full continuum recovery is named as out of scope.
What Lives in the Paper
Theorem-grade content
V600 setup, σ-pair definition, monochromatic spectrum theorem, per-coset first-law identity, PBH-style structural prediction at the discrete level.
Semiclassical sketch
A route from the discrete spectrum to a continuum analogue under a stated finite cutoff. Sketch only — not a recovery proof.
Continuum derivation
Planck-unit calibration, full semiclassical derivation of Hawking radiation, cosmology, τσ construction (Paper 3), QG-as-theory claims.
What Is Explicitly Out of Scope
- Planck-unit calibration of the 5/2 to physical units
- Full semiclassical derivation of Hawking continuum radiation
- Cosmology, expansion rates, CMB — this paper is purely group-theoretic
- The construction of the τσ involution (Paper 3 territory)
- Quantum-gravity-as-complete-theory claims
- Non-mobile-vertex excitations — the spectrum theorem is restricted to the 96 mobile vertices
Reviewer Risk Register
The paper enters its release with five anticipated objections each addressed by an explicit mitigation:
- "The delta-function spectrum is unphysical." Mitigation: it is a finite-group spectrum, not a continuum density. Continuum recovery is sketched, not claimed.
- "Why σ-pairs?" Mitigation: the σ-pairing is canonical — it is the orbit structure of the σ-Galois action on the mobile set. No alternative pairing is invoked.
- "What sets the cutoff?" Mitigation: the cutoff is explicit and the paper does not pretend to derive it from internal structure alone.
- "What is the role of W(H4)?" Mitigation: the underlying H4 Coxeter symmetry is recorded but not load-bearing in this theorem.
- "How strong is the PBH prediction?" Mitigation: it is a structural prediction at the discrete level — not a quantitative astrophysical claim.
Verification
The certificate verify.py enumerates all 48 σ-pairs across the 6 Dic5-cosets, computes the σ-pair observable in pure Python rationals, and confirms the eigenvalue 5/2 across every pair. The first-law identity is checked per coset against the Paper-1 incidence counts. No floating-point arithmetic.
One observable. Ninety-six vertices. One eigenvalue. The arithmetic is exact; the continuum question is open.